A survey on signature-based Gr\"obner basis computations

This paper is a survey on the area of signature-based Gr\"obner basis algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table

Essays on strategic trading

This dissertation discusses various aspects of strategic trading using both analytical modeling and numerical methods. Strategic trading, in short, encompasses models of trading, most notably models of optimal execution and portfolio selection, in which one seeks to rigorously consider various---both explicit and implicit---costs stemming from the act of trading itself. The strategic trading approach, rooted in the market microstructure literature, contrasts with many classical finance models in which markets are assumed to be frictionless and traders can, for the most part, take prices as given. Introducing trading costs to dynamic models of financial markets tend to complicate matters. First, the objectives of the traders become more nuanced since now overtrading leads to poor outcomes due to increased trading costs. Second, when trades affect prices and there are multiple traders in the market, the traders start to behave in a more calculated fashion, taking into account both their own objectives and the perceived actions of others. Acknowledging this strategic behavior is especially important when the traders are asymmetrically informed. These new features allow the models discussed to better reflect aspects real-world trading, for instance, intraday trading patterns, and enable one to ask and answer new questions, for instance, related to the interactions between different traders. To efficiently analyze the models put forth, numerical methods must be utilized. This is, as is to be expected, the price one must pay from added complexity. However, it also opens an opportunity to have a closer look at the numerical approaches themselves. This opportunity is capitalized on and various new and novel computational procedures influenced by the growing field of numerical real algebraic geometry are introduced and employed. These procedures are utilizable beyond the scope of this dissertation and enable one to sharpen the analysis of dynamic equilibrium models.Tämä väitöskirja käsittelee strategista kaupankäyntiä hyödyntäen sekä analyyttisiä että numeerisia menetelmiä. Strategisen kaupankäynnin mallit, erityisesti optimaalinen kauppojen toteutus ja portfolion valinta, pyrkivät tarkasti huomioimaan kaupankäynnistä itsestään aiheutuvat eksplisiittiset ja implisiittiset kustannukset. Tämä erottaa strategisen kaupankäynnin mallit klassisista kitkattomista malleista. Kustannusten huomioiminen rahoitusmarkkinoiden dynaamisessa tarkastelussa monimutkaistaa malleja. Ensinnäkin kaupankävijöiden tavoitteet muuttuvat hienovaraisemmiksi, koska liian aktiivinen kaupankäynti johtaa korkeisiin kaupankäyntikuluihin ja heikkoon tuottoon. Toiseksi oletus siitä, että kaupankävijöiden valitsemat toimet vaikuttavat hintoihin, johtaa pelikäyttäytymiseen silloin, kun markkinoilla on useampia kaupankävijöitä. Pelikäyttäytymisen huomioiminen on ensiarvoisen tärkeää, mikäli informaatio kaupankävijöiden kesken on asymmetristä. Näiden piirteiden johdosta tässä väitöskirjassa käsitellyt mallit mahdollistavat abstrahoitujen rahoitusmarkkinoiden aiempaa täsmällisemmän tarkastelun esimerkiksi päivänsisäisen kaupankäynnin osalta. Tämän lisäksi mallien avulla voidaan löytää vastauksia uusiin kysymyksiin, kuten esimerkiksi siihen, millaisia ovat kaupankävijöiden keskinäiset vuorovaikutussuhteet dynaamisilla markkinoilla. Monimutkaisten mallien analysointiin hyödynnetään numeerisia menetelmiä. Tämä avaa mahdollisuuden näiden menetelmien yksityiskohtaisempaan tarkasteluun, ja tätä mahdollisuutta hyödynnetään pohtimalla laskennallisia ratkaisuja tuoreesta numeerista reaalista algebrallista geometriaa hyödyntävästä näkökulmasta. Väitöskirjassa esitellyt uudet laskennalliset ratkaisut ovat laajalti hyödynnettävissä, ja niiden avulla on mahdollista terävöittää dynaamisten tasapainomallien analysointia

Joint shape and motion estimation from echo-based sensor data

2018 Fall.Includes bibliographical references.Given a set of time-series data collected from echo-based ranging sensors, we study the problem of jointly estimating the shape and motion of the target under observation when the sensor positions are also unknown. Using an approach first described by Stuff et al., we model the target as a point configuration in Euclidean space and estimate geometric invariants of the configuration. The geometric invariants allow us to estimate the target shape, from which we can estimate the motion of the target relative to the sensor position. This work will unify the various geometric- invariant based shape and motion estimation literature under a common framework, and extend that framework to include results for passive, bistatic sensor systems

Computational Methods for Computer Vision : Minimal Solvers and Convex Relaxations

Robust fitting of geometric models is a core problem in computer vision. The most common approach is to use a hypothesize-and-test framework, such as RANSAC. In these frameworks the model is estimated from as few measurements as possible, which minimizes the risk of selecting corrupted measurements. These estimation problems are called minimal problems, and they can often be formulated as systems of polynomial equations. In this thesis we present new methods for building so-called minimal solvers or polynomial solvers, which are specialized code for solving such systems. On several minimal problems we improve on the state-of-the-art both with respect to numerical stability and execution time.In many computer vision problems low rank matrices naturally occur. The rank can serve as a measure of model complexity and typically a low rank is desired. Optimization problems containing rank penalties or constraints are in general difficult. Recently convex relaxations, such as the nuclear norm, have been used to make these problems tractable. In this thesis we present new convex relaxations for rank-based optimization which avoid drawbacks of previous approaches and provide tighter relaxations. We evaluate our methods on a number of real and synthetic datasets and show state-of-the-art results

Satisfiability Modulo Finite Fields

We study satisfiability modulo the theory of finite fields and give a decision procedure for this theory. We implement our procedure for prime fields inside the cvc5 SMT solver. Using this theory, we con- struct SMT queries that encode translation validation for various zero knowledge proof compilers applied to Boolean computations. We evalu- ate our procedure on these benchmarks. Our experiments show that our implementation is superior to previous approaches (which encode field arithmetic using integers or bit-vectors)

Tropicalization of del Pezzo surfaces

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, respectively. There are 27 trees on each tropical cubic surface, attached to a bounded complex with up to 73 polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic types of such surfaces